Correlations and Two-Body Currrents in Weak Processes

Correlations and two-body currents in (electro)weak processes

Saori Pastore INT Program INT-17-2a - Neutrinoless Double-beta Decay Seattle WA - June 2017

WITH nnn Carlson & Gandolﬁ (LANL) - Schiavilla & Baronibla (ODU/JLAB) - Wiringa & Piarulli & Pieper (ANL) Mereghetti & Dekens & Cirigliano (LANL) REFERENCES nnn PRC78(2008)064002- PRC80(2009)034004 - PRL105(2010)232502 - PRC84(2011)024001 - PRC87(2013)014006 PRC87(2013)035503 - PRL111(2013)062502 - PRC90(2014)024321 - JPhysG41(2014)123002 - PRC(2016)015501

1/38 Fundamental Physics Quests: Double Beta Decay

observation of 0νββ-decay → lepton # L = l ¯l not conserved − implications→ in matter-antimatter imbalance Majorana Demonstrator

* detectors’ active material 76Ge * 25 10 0νββ-decay τ1/2 & 10 years (age of the universe 1.4 10 years) 1 ton of material to see (if any) 5 decays per× year * also, if nuclear m.e.’s are known, absolute∼ν-masses can be extracted *

2015 Long Range Plane for Nuclear Physics

2/38 The question

What are the present uncertainties in nuclear matrix elements relevant for • neutrinoless double beta decay, and how can they be improved?

OUTLINE Role of correlations and many-body currents in * Single beta-decay in A 10 nuclei * * Neutrinoless double beta-decay≤ in A 12 nuclei * ≤

3/38 Nuclear Interactions

The nucleus is made of A non-relativistic interacting nucleons and its energy is

A H = T + V = ∑ ti + ∑υij + ∑ Vijk + ... i=1 i

-20 1+ + + + + 2 2 7/2 + 4 + 0 + + 2 + 0 -30 0 3 5/2 + 2 + π + 0 + 1 4 1 5/2 2 + He 6 + 4 7/2+ He 7/2 8 1 + 4+ ∆ 6 1/2 He + 3 5/2+ -40 Li 0 + 5/2 2+ 3/2 + 1 7/2 3 + 1/2 3+ + 2 7/2 1 + 3/2 + 7 4 2 + -50 Li 2+ 3/2 + 3 9 + 1+ 3 + 2+ Li 3/2 + 4 N N 8 + + 1 + + 5/2 3,2 + 2 Li 0 + 1 1/2 0 3+ -60 Argonne v + 18 8 5/2 2 2+ Be + 1/2+ 2 1+ + with Illinois-7 3/2 0 1+ Energy (MeV) -70 GFMC Calculations 9 10 3+ * AV18+UIX / AV18+IL7 - QMC Be Be 10 -80 B 0+ AV18 2+ * NN(N3LO)+3N(N2LO) - QMC -90 AV18 Expt. 0+ +IL7 (π N ∆) by Maria Piarulli et al. 12 -100 C PRC91(2015)024003

Carlson et al. Rev.Mod.Phys.87(2015)1067

4/38 Correlations in our formalism

Minimize expectation value of H = T + AV18 + IL7

ΨV H ΨV EV = h | | i E0 ΨV ΨV ≥ h | i using trial function

S ΨV = ∏(1 + Uij + ∑ Uijk) ∏fc(rij) ΦA(JMTT3) | i " i

* single-particle ΦA(JMTT3) is fully antisymmetric and translationally invariant * central pair correlations fc(r) keep nucleons at favorable pair separation

* pair correlation operators Uij reﬂect inﬂuence of υij (AV18)

* triple correlation operators Uijk reﬂect the inﬂuence of Vijk (IL7)

In an uncorrelated wave function 1) Uij and Uijk are turned off, and 2) only the dominant spatial symmetry is kept

Lomnitz-Adler, Pandharipande, and Smith NPA361(1981)399 Wiringa, PRC43(1991)1585

5/38 Electroweak Reactions

* ω 102 MeV: Accelerator neutrinos * ω ∼ 101 MeV: EM decay, β-decay ∼ * ω . 101 MeV: Nuclear Rates for Astrophysics

ℓ′ µ e′ ,p′e P µ, Ψ f | f i q θe γ∗ √α Z√α jµ ℓ qµ = pµ p µ e − ′e = (ω, q) µ e,pµ P , Ψi e i | i

6/38 Nuclear Currents

1b 2b

ℓ A ′ ℓ′ ρ = ∑ ρi + ∑ρij + ... , i=1 i

* In Impulse Approximation IA nuclear currents are expressed in terms of those associated with individual protons and nucleons, i.e., ρi and ji , 1b-operators

L~ p

S~n S~p

* Two-body 2b currents essential to satisfy current conservation

q + . . . q j = [H, ρ ] = [ti + υij + Vijk, ρ ] γ ·

N N 7/38 Electromagnetic Currents from Nuclear Interactions (SNPA currents)

q j =[H, ρ ] = ti + υij + Vijk, ρ ·

1) Longitudinal component ﬁxed by current conservation 2) Plus transverse “phenomenological” terms

transverse j = j(1) π π ρω + j(2)(v) + ∆ + q

N N + j(3)(V )

Villars, Myiazawa (40-ies), Chemtob, Riska, Schiavilla . . . see, e.g., Marcucci et al. PRC72(2005)014001 and references therein

8/38 Currents from nuclear interactions

Satisfactory description of a variety of nuclear em properties in A 12 ≤

2H(p,γ)3He capture

0.5

0.4

0.3

S(E) (eV b) 0.2 LUNA Griffiths et al. Schmid et al. 0.1

0 0 10 20 30 40 50

ECM(keV)

Marcucci et al. PRC72, 014001 (2005)

9/38 Electromagnetic Currents from Chiral Effective Field Theory

LO : j( 2) eQ 2 − ∼ −

NLO : j( 1) eQ 1 − ∼ −

N2LO : j( 0) eQ0 − ∼

* 3 unknown Low Energy Constants: ﬁxed so as to reproduce d, 3H, and 3He magnetic moments

N3LO: j(1) eQ ∼

unknown LEC′s

Pastore et al. PRC78(2008)064002 & PRC80(2009)034004 & PRC84(2011)024001 * analogue expansion exists for the Axial nuclear current - Baroni et al. PRC93 (2016)015501 *

10/ 38 Electromagnetic LECs dV, dV S V V 1 2 d , d1 , d2 cS, cV Isovector

V d = 4µ hA/9mN (m mN) and dS, dV , and dV could be determined by 2 ∗ ∆ − 1 2 dV = 0.25 dV πγ-production data on the nucleon 1 × 2 assuming ∆-resonance saturation

Left with 3 LECs: Fixed in the A = 2 3 nucleons’ sector − * Isoscalar sector: S S 3 3 * d and c from EXPT µd and µS( H/ He) * Isovector sector: * cV from EXPT npdγ xsec. or V 3 3 * c from EXPT µV ( H/ He) m.m.

* Regulator C(Λ) = exp( (p/Λ)4) with Λ = 500 600 MeV − −

Λ NN/NNN 10 dS cS 500 AV18/UIX (N3LO/N2LO) –1.731× (2.190) 2.522 (4.072) 600 AV18/UIX (N3LO/N2LO) –2.033 (3.231) 5.238 (11.38) 11/ 38 Electromagnetic LECs dV, dV S V V 1 2 d , d1 , d2 cS, cV Isovector

V d = 4µ hA/9mN (m mN) and dS, dV , and dV could be determined by 2 ∗ ∆ − 1 2 dV = 0.25 dV πγ-production data on the nucleon 1 × 2 assuming ∆-resonance saturation

* Regulator C(Λ) = exp( (p/Λ)4) with Λ = 500 600 MeV − −

V V Λ NN/NNN Current d1 c 600 AV18/UIX I 4.98 -11.57 II 4.98 -1.025 12/ 38 Convergence and cutoff dependence

np capture x-section/ µV of A = 3 nuclei bands represent nuclear model dependence [NN(N3LO)+3N(N2LO) – AV18+UIX]

360 γ 3 3 -1.8 np µV( H/ He) 340 -2

320 -2.2 mb 300 n.m. LO -2.4 NLO 280 N2LO N3LO (no LECs) -2.6 N3LO (full) 260 EXP -2.8

500 600 Λ (MeV) Λ(MeV)

Piarulli et al. PRC(2013)014006

13/ 38 Calculations with EM Currents from χEFT with π’s and N’s

◮ Park, Min, and Rho et al. (1996) applications to A=2–4 systems by Song, Lazauskas, Park at al. (2009-2011) within the hybrid approach ...... * Based on EM χEFT currents from NPA596(1996)515 ◮ Meissner and Walzl (2001); K¨olling, Epelbaum, Krebs, and Meissner (2009–2011) applications to: d and 3He photodisintegration by Rozpedzik et al. (2011); e-scattering (2014); d magnetic f.f. by K¨olling, Epelbaum, Phillips (2012); radiative N d capture by Skibinski et al. (2014) − ...... * Based on EM χEFT currents from PRC80(2009)045502 & PRC84(2011)054008 and consistent χEFT potentials from UT method ◮ Phillips (2003-2007) applications to deuteron static properties and f.f.’s * ......

14/ 38 Magnetic Moments and M1 Transitions

9 5 - 3 - Be( /2 /2 ) B(E2)

4 9 5 - 3 - Be( /2 /2 ) B(M1)

3 8B(3+ 2+) B(M1) 7 9 Li B 8 + + p 3 9Li B(1 2 ) B(M1) 2 H 8Li(3+ 2+) B(M1) 6Li* 10B

) 1 8 8Li(1+ 2+) B(M1) N Li 8B µ 2H 6Li ( 10 B* 7 1 - 3 - µ Be( / / ) B(M1) 0 GFMC(1b) 2 2 9C GFMC(1b+2b) 9 7Li(1/ - 3/ -) B(E2) 7Be Be 2 2 -1 EXPT 7Li(1/ - 3/ -) B(M1) n 3He 2 2 -2 6Li(0+ 1+) B(M1)

-3 EXPT GFMC(1b) GFMC(1b+2b) 0 1 2 3 Ratio to experiment * 2b electromagnetic currents bring the THEORY in agreement with the EXPT * 40% 2b-current contribution found in 9C m.m. ∼ * 60 70% of total 2b-current component is due to one-pion-exchange currents ∼ − * 20-30% 2b found in M1 transitions in 8Be ∼

Pastore et al. PRC87(2013)035503 & PRC90(2014)024321, Datar et al. PRL111(2013)062502

15/ 38 Error Estimate

4 EE et al. error algorithm Epelbaum, Krebs, and 3 9 Meissner EPJA51(2015)53 7Li B p 3 9Li 2 H 6Li* 10 B δ N3LO =max Q4µLO, Q3 µLO µNLO ,

) 1 8 | − | N Li 8B µ 2H 6Li 2 NLO N2LO ( 10B* Q µ µ , µ 0 GFMC(IA) | − | 9 GFMC(FULL) C 1 N2LO N3LO 7 9Be Q µ µ EXPT Be | − | -1 n 3He m p -2 Q = max π , Λ Λ -3

m.m. THEO EXP 9C -1.35(4)(7) -1.3914(5) 9Li 3.36(4)(8) 3.4391(6)

* ‘N3LO-∆’ corrections can be ’large’ * * SNPA and χEFT currents qualitatively in agreement, χEFT isoscalar currents provide better description exp data * Pastore et al. PRC87(2013)035503

16/ 38 Two-body M1 transitions densities

0.04 0.04 0.04 Total 0.03 0.03 NLO-OPE0.03 )

-3 N2LO-RC 0.02 0.02 N3LO-CT0.02 fm

N N3LO-TPE µ 0.01 0.01 N3LO-∆ 0.01

(r) ( 0.00 0.00 0.00 M1 ρ -0.01 8 + + -0.01 8 + + -0.01 8 + + Be(1 ;122 ;0) Be(1 ;02 ;1) Be(1 ;10 ;0) -0.020.02 -0.020.02 -0.020.02 0 1 2 30 1 2 30 1 2 3 4 Total )

-3 N2LO-RC 0.01 0.01 0.01 fm N3LO-CT N µ N3LO-ρπγ

(r) ( 0.00 0.00 0.00 M1 ρ 8 + + 8 + + 8 + + Be(1 ;12 ;1) Be(1 ;022 ;0) Be(1 ;02 ;0) -0.01 -0.01 -0.01 0 1 2 30 1 2 30 1 2 3 4

r12 (fm) r12 (fm) r12 (fm)

(J ,T ) (J ,T ) IA NLO-OPE N2LO-RC N3LO-TPE N3LO-CT N3LO-∆ MEC i i → f f (1+;1) (2+;0) 2.461 (13) 0.457 (3) -0.058 (1) 0.095 (2) -0.035 (3) 0.161 (21) 0.620 (5) → 2

Pastore et al. PRC90(2014)024321

17/ 38 β decay −

The “gA problem” and the role of two-nucleon correlations and two-body currents

e−

ν¯e

gA nucleon axial coupling constant

Berna U.

*Preliminary results*

18/ 38 Theory vs Experiment: The “gA problem”

eff g 0.70gA A ≃

Fig. from Chou et al. PRC47(1993)163

19/ 38 Correlations in our formalism

Minimize expectation value of H = T + AV18 + IL7

ΨV H ΨV EV = h | | i E0 ΨV ΨV ≥ h | i using trial function

S ΨV = ∏(1 + Uij + ∑ Uijk) ∏fc(rij) ΦA(JMTT3) | i " i

* single-particle ΦA(JMTT3) is fully antisymmetric and translationally invariant * central pair correlations fc(r) keep nucleons at favorable pair separation

* pair correlation operators Uij reﬂect inﬂuence of υij (AV18)

* triple correlation operators Uijk reﬂect the inﬂuence of Vijk (IL7)

In an uncorrelated wave function 1) Uij and Uijk are turned off, and 2) only the dominant spatial symmetry is kept

Lomnitz-Adler, Pandharipande, and Smith NPA361(1981)399 Wiringa, PRC43(1991)1585

20/ 38 Role of correlations in beta-decay m.e.’s

10 10 C B ; q = 0.76

7 7 Be Li ; q = 0.87

6 6 He Li ; q = 0.82

3 3 H He ; q = 0.97

Ratio to EXPT Simple w.f. (1b)

1 1.2 1.4 1.6 1.8

q = quenching from correlations data from TUNL compilations & Suzuki et al. PRC67(2003)044302 & Chou et al. PRC47(1993)163 * Preliminary *

21/ 38 SNPA Two-body Axial Currents

1) One body has GT, relativistic corrections, PS from pion-pole diagrams 2) Two-body currents 2.a) Major contribution from ∆-excitation current 2.b) Negligible contributions from Aπ, Aρ, Aπρ 3) AN∆ coupling ﬁxed to tritium beta-decay 4) 3% additive correction from ∆-current ∼ Chemtob, Rho, Towner, Riska, Schiavilla, Marcucci ... see, e.g., Marcucci et al. PRC63(2001)015801 and references therein

22/ 38 Two-body Axial Currents from χEFT

c3 and c4 * are saturated by the ∆ and ρπ d.o.f. * enter also the χEFT two- and three-nucleon χEFT potential * are taken them from Entem and Machleidt c = 3.2 GeV 1, c = 5.4 GeV 1 3 − − 4 − PRC68(2003)041001 & Phys.Rep.503(2011)1

A. Baroni et al. PRC93(2016)015501 & PRC94(2016)024003

23/ 38 Two-body Axial Currents from χEFT

cD * ﬁtted to GT m.e. of tritium beta-decay * for both χEFT potentials and AV18+UIX

* because of N4LO two-body currents cD value changes

N3LO N4LO Λ 500 600 500 600 cD –0.353 –0.443 –1.847 –2.030

A. Baroni et al. PRC93(2016)015501 & PRC94(2016)024003

24/ 38 Three-body Axial Currents from χEFT

A. Baroni et al. PRC93(2016)015501 & PRC94(2016)024003

25/ 38 Convergence and cutoff dependence

Tritium β-decay

1.08

GT 1.06 + N2LO (1b RC)

+ N3LO (OPE - c3 c4) + N3LO (CT c ) 1.04 D + N4LO (OPE) + N4LO (MPE) 1.02 + N4LO (3b)

1 EXP Cumulative m.e. / EXP

0.98 GT (LO)

0.96 500 600 700 Cutoff MeV

* 2% additive contribution from two-body currents ∼

A. Baroni et al. PRC93(2016)015501 & PRC94(2016)024003

26/ 38 Calculations with EW Currents from χEFT with π’s and N’s

Incomplete history

◮ Park, Min, and Rho et al. (90-ies) applications to A=2–4 systems including µ-capture, pp-fusion, hep · ◮ Krebs and Epelbaum et al. (2016) ◮ Klos et al. (2015) * ......

27/ 38 Role of two-body currents in beta-decay m.e.’s

SNPA currents χEFT currents VMC Calculations GFMC calculations

10 10 10 10 C B Coming soon C B

7 7 7 7 Be Li Be Li

6 6 6 6 He Li He Li

3 3 3 3 H He H He

Correlated w.f. (1b+2b) gfmc (1b) Ratio to EXPT Correlated w.f. (1b) Ratio to EXPT gfmc (1b+2b) Simple w.f. (1b) 1 1.2 1.4 1.6 1 1.2 1.4 1.6

*Preliminary* * SNPA and χEFT two-body currents are qualitatively in agreement (both are ﬁtted to the tritium β-decay) * Two-body currents are found to provide a small (negligible) contribution to the

quenching, limited to the light systems we studied 28/ 38 χEFT currents: a closer look

A = 7 Captures

gs ex LO 2.334 2.150 2 2 N2LO –3.18 10− –2.79 10− × 2 × 2 N3LO(OPE) –2.99 10− –2.44 10− × 1 × 1 N3LO(CT) 2.79 10− 2.36 10− × 1 × 1 N4LO(2b) –1.61 10− –1.33 10− N4LO(3b) –6.59×10 3 –4.86×10 3 × − × − TOT(2b+3b) 0.050 0.046

* Large cancellations due to positive CT at N3LO with cD ﬁxed to GT m.e. of tritium

In preparation

29/ 38 ββ decay −

The “gA problem” and the role of two-nucleon correlations and two-body currents

e−

ν¯e

gA nucleon axial coupling constant

Berna U.

*Preliminary results*

30/ 38 Double beta-decay m.e.’s: Correlations

*Preliminary*

8He(0+;2) -> 8Be(0+;0) - AV18+UX 0.008

+ + 0.004 ρV = τ 1τ 2 /r12 Jf|ρV|Ji = .0406

f|ρV|i = .0227

0.000 ) -3

-0.004 (r) (fm 12 ρ + + -0.008 ρA = 1•2 τ 1τ 2 /r12 f|ρA|i = -.0368

Jf|ρA|Ji = -.122

-0.012

-0.016 0 1 2 3 4 5 r (fm)

* < ρV >corr 0.56 < ρV >uncorr , qV = 0.75 ∼ * < ρA >corr 0.30 < ρA >uncorr , qA = 0.55 ∼ Bob Wiringa et al.

31/ 38 Double beta-decay m.e.’s: Correlations

*Preliminary*

12Be(0+;2) -> 12C(0+;0) - AV18+UX 0.024

+ + 0.016 ρV = τ 1τ 2 /r12 Jf|ρV|Ji = .063

0.008 f|ρV|i = .0545

0.000 ) -4 -0.008 (r) (fm

12 -0.016 f|ρA|i = -.137 ρ + + -0.024 ρA = 1•2 τ 1τ 2 /r12 Jf|ρA|Ji = -.19

-0.032

-0.040 0 1 2 3 4 5 r (fm)

* < ρV >corr 0.86 < ρV >uncorr , qV = 0.93 ∼ * < ρA >corr 0.72 < ρA >uncorr , qA = 0.85 ∼ Bob Wiringa et al.

32/ 38 Double beta-decay m.e.’s: Two-body currents

π ππ CT

σ1 σ2 υst = Lst τ1,+ τ2,+ · 2 mπ q σ qσ q υ = L τ τ 1 2 ππ ππ 1,+ 2,+ · 2 ·2 2 mπ (q + mπ ) σ qσ q υ = L τ τ 1 2 π π 1,+ 2,+ 3 · 2 · 2 mπ (q + mπ ) σ1 σ2 υCT = LCT τ1,+ τ2,+ ·3 mπ

Lππ , Lπ , LCT are model dependent

WITH Emanuele Mereghetti & Dekens & Cirigliano & Graesser & Wiringa et al.

33/ 38 Double beta-decay m.e.’s in 6He(0+;2) 6Be(0+;0): A test case I →

-3 1.0 10

+ +σ σ 0.0 Axial ∝ τ1 τ2 1 2 + + · Tensor ∝ τ1 τ2 S12

] -3

(-1) -1.0 10

-3 Density [fm -2.0 10 Axial Standard = -0.1105 ππ Axial = 0.0042 ππ Tensor = 0.0003 -3 * Preliminary * -3.0 10 π Axial = -0.0093 π Tensor = 0.0013

-3 -4.0 10 0 2 4 6 8 r [fm]

π ππ CT WITH Emanuele Mereghetti & Dekens & Cirigliano & Graesser & Wiringa et al.

34/ 38 Double beta-decay m.e.’s in 6He(0+;2) 6Be(0+;0): A test case I →

-3 2.0 10 + + Axial ∝ τ τ σ1 σ2 -3 1 2 1.0 10 + + · Tensor ∝ τ1 τ2 S12

0.0 2 ] e (r/R) (-1) C(r) = − -3 3/2 3 -1.0 10 (π) R Density [fm -3 -2.0 10 Axial Stndard = -0.1105 CT Axial (R=0.5) = -0.0119

-3 CT Fermi (R=0.5) = 0.0040 -3.0 10 * Preliminary *

-3 -4.0 10 0 2 4 6 2mπ mπ r [fm]

π ππ CT

WITH Emanuele Mereghetti & Dekens & Cirigliano & Graesser & Wiringa et al.

35/ 38 Double beta-decay m.e.’s in 8He(0+;2) 8Be(0+;0): A test case II →

-4 4.0 10

-4 2.0 10 + +σ σ Axial ∝ τ1 τ2 1 2 0.0 + + · Tensor ∝ τ1 τ2 S12

] -4

(-1) -2.0 10

-4 -4.0 10 Standard Axial = -0.0044 Tensor = 0.0018

Density [fm π -4 ππ Tensor = 0.0003 -6.0 10 ππ Axial = 0.0011 π Axial = -0.0016 -4 * Preliminary * -8.0 10

-3 -1.0 10

0 2 4 6 r [fm]

π ππ CT

WITH Emanuele Mereghetti & Dekens & Cirigliano & Graesser & Wiringa et al.

36/ 38 Double beta-decay m.e.’s in 8He(0+;2) 8Be(0+;0): A test case II →

-4 + + 4.0 10 Axial ∝ τ τ σ σ 1 2 1 · 2 -4 + + 2.0 10 Tensor ∝ τ1 τ2 S12 0.0 2 ] (r/R) (-1) -4 e− -2.0 10 C(r) = 3/2 3 -4 (π) R -4.0 10 Standard Axial = -0.0044

Density [fm CT Fermi (R=0.5) = 0.0012 -4 -6.0 10 CT Fermi (R=0.8) = 0.0018

-4 CT Axial (R=0.5) = -0.0037 -8.0 10 CT Axial (R=0.8) = -0.0054

-3 -1.0 10 * Preliminary *

-3 -1.2 10 0m 2 4 6 π r [fm]

π ππ CT WITH Emanuele Mereghetti & Dekens & Cirigliano & Graesser & Wiringa et al.

37/ 38 Summary and Outlook

We discussed the role played by correlations and many-body currents in β- and υ0ββ-decay m.e.’s of A 12 nuclei ≤ * Two-body currents (both SNPA and χEFT) provide negligible quenching in the β-decay m.e.’s we studied

* Correlations provide a quenching q 0.95 in A = 3 and q 0.76 in A = 10 β-decay m.e.’s ∼ ∼

* Correlations affect υ0ββ-decay m.e.’s leading to a quenching q 0.55 in ∼ Standard Axial A = 8 and q 0.93 in Standard Fermi A = 12 ∼ * A cancellation in the Axial Standard two-body current in A = 8 υ0ββ-decay m.e.’s could enhance contributions from Non-Standard two-body currents

Outlook

* Benchmark both single- and double-beta decay m.e.’s * Characterize two-body currents entering double-beta decay m.e.’s * Calculate more single- and double-beta decay m.e.’s and study model dependence using AV18+IL7 and ∆-full chiral potential by Piarulli et al.

38/ 38